Differential calculus limits

By analogy with ordinary differential calculus, the ratio du/df is defined as the limit of the ratio as the interval t becomes progressively smaller. 

Sections 2.1 and 2.2). However, the limit 0f this ratio may exist In fact this principle is the very basis of the differential calculus, as indicated by Eq. (9). 

The validity of (1.3), i.e., the existence of the derivative dz/dx in (1.4), is an essential requisite for application of the formalism of differential calculus. It is therefore important that the magnitudes of differentials dz, dx be taken sufficiently small (but not zero, a meaningless and unphysical extrapolation in this context ) for the limiting ratio in (1.4) to have an experimentally well-defined value, within usual limits of experimental precision. 

The solution to the general problem of determining the area under a curve arises directly from differential calculus, the concept of limits, and the infinitesimal. Seventeenth century mathematicians began to think of the area, not as a whole, but as made up of a series of rectangles, of width Ax, placed side by side, and which, together, cover the interval [a,b (see Figure 6.2). 

The concept of limit seems to be essential in the understanding and the present teaching of Calculus. In this article, however, we show how to structure and use differential calculus without introducing this concept. The crucial idea in this development is to use Leibniz rule for the derivative of a product of two functions as one of the postulates, rather than as a derived theorem. Within this approach, the idea of limit could be introduced belatedly and only in order to define concepts such as continuity and differentiability in a more rigorous fashion. 

Caveat The purpose of this paper is to introduce the principles on how to teach differential calculus without the idea of a limit. A more pedagogical version is being prepared. 

These three rules define the differential calculus without involving the concept of limit. The derivative of positive powers and of polynomials follow directly from a straightforward application of these rules. The case of the... 

Application of the mean value theorems of integral and differential calculus followed by the limiting process wherein AZ and AW are allowed to go to zero yields... 

Application of the mean value theorem of differential calculus to the first two terms and the mean value theorem of integral calculus to each of the integrals of Eq. (13-82) followed by the limiting process whereby AZ is allowed to go to zero yields the following result upon recognition that Z and Z + AZ were selected arbitrarily in the domain (0 < Z < ZT) of interest... 

Note the equivocal use of the word limit. There is a difference between the limit of the differential calculus and the limit of the integral calculus. 

In the notation of differential calculus, this limit is written as... 

One speaks of taking a limit of a function /(x) as X approaches a particular value, for example, x= a. This means that the function is examined on an interval around, but not including x= a. Values of /(x) are taken on that interval as the varying x values get closer and closer to the target value of x = functional values is examined as x approaches a. If those values continue to approach a single target value, it is that value that is said to be equal to the limit of (x) as x approaches a. Otherwise, the limit is said not to exist This method is used in both differential calculus and integral calculus. 

It is known from differential calculus that the integral of a function f(x) is equivalent to the area between the function and the x axis enclosed within the limits of integration, as shown in Fig. 4. Ifl. Any portion of the area that is below the x axis is counted as negative area (Fig. A.lb). Therefore, one way of evaluating the integral... 

The mathematical knowledge pre-supposed is limited to the elements of the differential and integral calculus for the use of those readers who possess my Higher Mathematics for... 

This is an introductory book. The pace is leisurely, and where needed, time is taken to consider why certain assumptions are made, to discuss why an alternative approach is not used, and to indicate the limitations of the treatment when applied to real situations. Although the mathematical level is not particularly difficult (elementary calculus and the linear first-order differential equation is all that is needed), this does not mean that the ideas and concepts being taught are particularly simple. To develop new ways of thinking and new intuitions is not easy. 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

This chapter introduces the concept of the limit, with a view not only to probing limiting behaviour of functions but also as a foundation to the development of differential and integral calculus in the following chapters. The key points discussed include ... 

We will discuss quantum mechanics extensively in Chapters 5 and 6. It provides the best description we have to date of the behavior of atoms and molecules. The Schrodinger equation, which is the fundamental defining equation of quantum mechanics (it is as central to quantum mechanics as Newton s laws are to the motions of particles), is a differential equation that involves a second derivative. In fact, while Newton s laws can be understood in some simple limits without calculus (for example, if a particle starts atx = 0 and moves with constant velocity vx,x = vxt at later times), it is very difficult to use quantum mechanics in any quantitative way without using derivatives. 

In a search problem, almost nothing is known in advance about how the criterion of effectiveness depends upon the operating variables, the only way to learn being to perform experiments. Here the obstacle to using the calculus is the complete lack of a function that can be differentiated. The objective of the search is to get as close as possible to the optimum after only a limited number of experiments. Box and Wilson, with their paper The Experimental Attainment of Optimum Conditions published in 1951, were the first to interest engineers in search problems (B4). 

In mathematics the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a variable quantity approaches as closely as one desires. The operations of differentiation and integration from calculus are both based on the theory of limits. The theory of limits is based on a particular property of the real numbers namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more. 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

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